Understanding Number Analogy Questions

This question is a type of number analogy where you need to find the relationship or pattern between pairs of numbers. The structure is given as A : B :: C : D :: E : F. This means that A is related to B in the same way that C is related to D, and E is related to F. We are given the pairs 396 : 24 and 672 : 30, and a partial pair ? : 28. We need to find the missing number represented by the question mark.

Analyzing the Given Number Pairs and Finding the Pattern

Let's look at the two complete pairs: 396 : 24 and 672 : 30. We need to find a consistent mathematical or logical pattern that connects the first number in each pair to the second number.

Let's break down the first number in each pair into its digits. For a three-digit number XYZ, the digits are X, Y, and Z.

• For 396 : 24, the number is 396. The digits are D1=3, D2=9, D3=6. The result is 24.
• For 672 : 30, the number is 672. The digits are D1=6, D2=7, D3=2. The result is 30.

Let's try applying various operations on the digits to see if we can arrive at the result.

Consider the product of the first two digits (D1 \(\times\) D2):

• For 396: D1 \(\times\) D2 = 3 \(\times\) 9 = 27. The result is 24. The difference is \(27 - 24 = 3\). Notice that 3 is the first digit (D1). So, \(27 - 3 = 24\). This fits the pattern \((D1 \times D2) - D1 = \text{Result}\).
• For 672: D1 \(\times\) D2 = 6 \(\times\) 7 = 42. The result is 30. The difference is \(42 - 30 = 12\). Notice that 12 is the product of the first and third digit (D1 \(\times\) D3 = 6 \(\times\) 2 = 12). So, \(42 - 12 = 30\). This fits the pattern \((D1 \times D2) - (D1 \times D3) = \text{Result}\).

We observe that the pattern involves calculating the product of the first two digits and then subtracting a value derived from the digits. However, the subtracted value seems to be different for the two given pairs: D1 for the first pair, and D1 \(\times\) D3 for the third pair (672 : 30 is the third pair in the analogy sequence A:B :: C:D :: E:F).

This suggests a pattern where the rule for subtraction depends on the position of the pair in the sequence. Let's hypothesize the following position-based rules:

• For the first pair (A:B): Result = (D1 \(\times\) D2) - D1
• For the second pair (C:D): Result = (D1 \(\times\) D2) - D2
• For the third pair (E:F): Result = (D1 \(\times\) D2) - (D1 \(\times\) D3)

Let's verify these rules with the given pairs:

• Pair 1 (396 : 24): Digits are 3, 9, 6. Applying the first rule: \((3 \times 9) - 3 = 27 - 3 = 24\). This matches the given result.
• Pair 3 (672 : 30): Digits are 6, 7, 2. Applying the third rule: \((6 \times 7) - (6 \times 2) = 42 - 12 = 30\). This matches the given result.

The rules seem consistent for the given pairs based on their position in the analogy sequence.

Finding the Missing Number using the Pattern

The missing number is in the second pair, ? : 28. Let the missing number be XYZ, with digits D1=X, D2=Y, D3=Z. According to our hypothesized position-based rules, the rule for the second pair is:

Result = (D1 \(\times\) D2) - D2

We are given that the result for this pair is 28. So, we need to find an option number XYZ such that:

\((D1 \times D2) - D2 = 28\)

Let's test the given options:

• Option 1: 504
  Digits are 5, 0, 4. D1=5, D2=0, D3=4.
  Calculate: \((5 \times 0) - 0 = 0 - 0 = 0\). This is not 28.
• Option 2: 536
  Digits are 5, 3, 6. D1=5, D2=3, D3=6.
  Calculate: \((5 \times 3) - 3 = 15 - 3 = 12\). This is not 28.
• Option 3: 588
  Digits are 5, 8, 8. D1=5, D2=8, D3=8.
  Calculate: \((5 \times 8) - 8 = 40 - 8 = 32\). This is not 28.
• Option 4: 572
  Digits are 5, 7, 2. D1=5, D2=7, D3=2.
  Calculate: \((5 \times 7) - 7 = 35 - 7 = 28\). This matches the required result.

Thus, the number 572 follows the pattern determined for the second pair in the analogy sequence.

Conclusion

The number related to 28 in the same way as 396 is related to 24 and 672 is related to 30 is 572. The pattern depends on the position of the number pair in the analogy: for the first pair, subtract the first digit from the product of the first two digits; for the second pair, subtract the second digit from the product of the first two digits; and for the third pair, subtract the product of the first and third digits from the product of the first two digits.

The final answer is 572.

Revision Table: Number Analogy Pattern

Additional Information on Logical Reasoning and Number Patterns

Number analogy questions are common in logical reasoning tests. They assess your ability to identify patterns and relationships between numbers. These patterns can be based on various properties and operations, including:

• Basic arithmetic operations (addition, subtraction, multiplication, division).
• Operations involving squares, cubes, or other powers.
• Operations based on the digits of the number (sum of digits, product of digits, operations on specific digits like first, last, or middle).
• Positional patterns, as seen in this question, where the rule applied might depend on the position of the number in the sequence or series.
• Prime numbers, composite numbers, or other number classifications.
• Sequences or series with a specific progression rule.

Solving these problems often requires systematic testing of different possible patterns and careful observation of the properties of the given numbers and results. Practice with various types of number pattern problems can help improve your ability to quickly identify the underlying logic.